“Teach A Level Maths” Vol. 1: AS Core Modules 6: Roots, Surds and Discriminant © Christine Crisp

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Let’s see why the answers (1) and (2) are the same Roots of Equations Roots is just another word for solutions ! e.g. Find the roots of the equation Solution: There are no factors, so we can either complete the square or use the quadratic formula. Completing the square: Using the formula: Let’s see why the answers (1) and (2) are the same

The answers from the quadratic formula can be simplified: We have Numbers such as are called surds However, 4 is a perfect square so can be square-rooted, so We have simplified the surd So, 2 is a common factor of the numerator, so

Exercise Simplify the following surds:

The Discriminant of a Quadratic Function The formula for solving a quadratic equation is The part is called the discriminant Because we square root the discriminant, we get different types of roots depending on its sign.

The Discriminant of a Quadratic Function we consider the graph of the function To investigate the roots of the equation The roots of the equation are at the points where y = 0 ( x = 1 and x = 4) The discriminant The roots are real and distinct. ( different )

The Discriminant of a Quadratic Function For the equation the discriminant The roots are real and equal ( x = 2)

The Discriminant of a Quadratic Function For the equation . . . . . . the discriminant There are no real roots as the function is never equal to zero If we try to solve , we get The square of any real number is positive so there are no real solutions to

SUMMARY The formula for solving the quadratic equation is The part is called the discriminant The roots are real and distinct ( different ) The roots are real and equal The roots are not real If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

Exercise 1 (a) Use the discriminant to determine the nature of the roots of the following quadratic equations: (i) (ii) (b) Check your answers by completing the square to find the vertex of the function and sketching. Solution: (a) (i) The roots are real and equal. (ii) The roots are real and distinct.

(b) Check your answers by completing the square to find the vertex of the function and sketching. (b) (i)  Vertex is ( -1,0 ) Roots of equation (real and equal) (ii)  Vertex is ( 1,-2 ) Roots of equation (real and distinct)

2. Determine the nature of the roots of the following quadratic equations ( real and distinct or real and equal or not real ) by using the discriminant. DON’T solve the equations. (a) Roots are real and equal (b) There are no real roots (c) Roots are real and distinct

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Roots is just another word for solutions ! e.g. Find the roots of the equation Completing the square: Solution: There are no factors, so we can either complete the square or use the quadratic formula. Using the formula: (1) (2)

The answers from the quadratic formula can be simplified: We have However, 4 is a perfect square so can be square-rooted, so So, 2 is a common factor of the numerator, so Numbers such as are called surds We have simplified the surd

The part is called the discriminant The roots are real and equal. The roots are not real. The roots are real and distinct. ( different ) The Discriminant The formula for solving the quadratic equation is If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

The roots are real and distinct. ( different ) The discriminant e.g. For

The discriminant e.g. For The roots are real and equal.

There are no real roots as the function is never equal to zero. The discriminant e.g. For